Séminaire Lotharingien de Combinatoire B36g (5pp.)

Plethysm and the Non-Compact Groups Sp(2n,R)

B.G.Wybourne

Instytut Fizyki, Uniwersytet Mikołaja Kopernika ul. Grudzia̧dzka 5/7 87-100 Toruń Poland

ABSTRACT Some preliminary results on the plethysms for the non-compact group Sp(2n, R) are presented. Complete results are given for power 2 plethysms of the two fundamental irreducible representations of Sp(2n, R). Several new S−function identities arise from this work. The stabilisation properties of the plethysms are briefly considered and some remarkable conjugacy mappings observed.

Introduction

The plethysm of S−functions has been the subject of much research ever since its introduction by Littlewood^{1}. Many applications have been made to the classical compact Lie groups by expressing the characters of the irreducible representations of the group in terms of S−functions^{2−6}. To date rather scant attention has been paid to application of plethysm to non-compact Lie groups^{7−9}.

The non-compact group Sp(2n, R) is of special interest in physics as it is the dynamical group^{10} of the n−dimensional isotropic harmonic oscillator which finds important applications in symplectic models of nuclei^{11} and in the mesoscopic physics of quantum dots^{12,13}. The non-trivial unitary irreducible representations of Sp(2n, R) are all of infinite dimension^{14,15}. An extensive outline of notation, characters, Kronecker products and branching rules is developed in reference 15. In other matters we follow the notation of Macdonald^{16} Arbitrary positive discrete harmonic series irreducible representations of Sp(2n, R) will be labelled as < \frac{k}{2} ; (λ) > or equivalently as < sκ; (λ) > where κ and s are the integer and residue parts of \frac{k}{2} .

The infinite set of states of a harmonic oscillator span the pair of infinite- dimensional fundamental unitary irreducible representations of Sp(2n, R) which we shall designate^{15} as < \frac{1}{2}; (0) > and < \frac{1}{2}; (1) >. Our central problem is to resolve symmetrised powers of these two irreducible representations which amounts to evaluating the plethysms

s_{λ}(< s; (0) >)  and  s_{λ}(< s; (1) >)  (1)

To proceed with the plethysm problem for Sp(2n, R) we first consider the Sp(2n, R) → U(n) decompositions where U(n) is the unitary group in n−dimensions and then show how these can be used to build up Sp(2n, R) plethysms up to some finite cutoff and present some complete results for λ ⊢ 2. We are then led to some new S−function identities and after some comments on the stability of Sp(2n, R) plethysms we end with a remarkable observation of the existence of a mapping between the two types of plethysms.

Sp(2n,R) → U(n) decompositions

Under the restriction^{15} Sp(2n, R) → U(n) a given irreducible representation of Sp(2n, R) decomposes into an infinite set of finite dimension irreducible representations of the unitary group U(n). In the case of the two fundamental irreducible representations of Sp(2n, R) we have^{15}

< s; (0) > → ε^{\frac{1}{2}} M_{+}  (2a)
< s; (1) > → ε^{\frac{1}{2}} M_{−}  (2b)

where M_{+} and M_{−} are the even and odd weight S−functions s_{m} appearing in the infinite series

M = ∑_{m=0}^{∞} s_{m}  (3)

In general one has^{15}

< \frac{k}{2} ; (λ) >→ ε^{\frac{k}{2}} · ((s_{λ_{s}})_{N}^{k} · D_{N})_{N}  (4)

where N = min(n, k) and D is the infinite S−function series

D = ∑_{δ} s_{δ}  (5)

where the δ are partitions involving only even parts. The subscript N means that all terms involving partitions into more than N parts are to be discarded. The first · indicates a product in U(n) and the second · a product in U(N). (s_{λ_{s}})^{k} is a signed sequence^{14,15} of terms ±s_{ρ} such that ±s_{ρ} is equivalent to s_{λ} under the modification rules of the orthogonal group O(k).

Plethysms in Sp(2n,R)

We are primarily interested in plethysms of the form s_{λ}(< s; (0) >) and s_{λ}(< s; (1) >). No general procedure seems to be known for evaluating Sp(2n, R) plethysms. Here we evaluate the terms, up to a given weight, by first decomposing the Sp(2n, R) irreducible representation into those of U(n), performing the plethysm at the U(n) level and then inverting to get irreducible representations of Sp(2n, R). This has been done for all λ ⊢ 4 and in some cases to λ ⊢ 6. Tables of the relevant plethysms are located at http://www.phys.uni.torun.pl/∼bgw/. In the case of λ ⊢ 2 it is possible to obtain completely general results as follows

s_{2}(< s; (0) >) = ∑_{i=0}^{∞} < 1; (0 + 4i) >

s_{1^{2}}(< s; (0) >) = ∑_{i=0}^{∞} < 1; (2 + 4i) >

s_{2}(< s; (1) >) = ∑_{i=0}^{∞} < 1; (2 + 4i) >

s_{1^{2}}(< s; (1) >) =< 1; (1^{2}) > + ∑_{i=0}^{∞} < 1; (4 + 4i) >  (6)

These results imply that the following S−function identity must hold

s_{1^{2}}(M_{+}) = s_{2}(M_{−})  (7)

as indeed may be shown to be the case^{17}.

If L_{+} and L_{−} are respectively the positive and negative terms of the series

L = ∑_{m=0}^{∞}(−1)^{m}s_{m}  (7)

then one finds

s_{1^{2}}(L_{+}) = s_{2}(L_{−})  (8)

Still further identities arise for the infinite S−function series defined by

A_{±} = L_{±}(s_{1^{2}})  B_{±} = M_{±}(s_{1^{2}})
C_{±} = L_{±}(s_{2})  D_{±} = M_{±}(s_{2})  (8)

Use of the associativety property of plethysms leads directly to

s_{1^{2}}(Z_{+}) = s_{2}(Z_{−})  (9)

for Z = A, B, C, D. Furthermore

s_{2}(Z) = ZZ_{+}  and  s_{1^{2}} = ZZ_{−}  (10)

The study of plethysms within the group Sp(2n, R) leads to still further identities. The observation that

s_{21^{2}}(< s; (0) >) = s_{31}(< s; (1) >)  (11)

leads to the remarkable S−function identity

s_{21^{2}}(M_{+}) = s_{31}(M_{−})  (12)

which generalises to

s_{σ}(s_{1^{2}}(M_{+})) = s_{σ}(s_{2}(M_{−}))  (13)

Again these identities extend to the series Z defined earlier.

Stability of Kronecker products and plethysms

A given plethysm, Kronecker product or decomposition will be said to be stable if at the stable value of n = n_{s} there is a one-to-one mapping between the resultant list of irreducible representations obtained at the stable value n_{s} and those obtained for all values of n > n_{s}. The Sp(2n, R) Kronecker product^{15}

< \frac{k}{2} (λ) > × < \frac{ℓ}{2}(ν) >=< \frac{(k + ℓ)}{2}; ((s_{λ_{s}})^{k} · (s_{ν_{s}})^{ℓ} · D)_{k+ℓ,n} >  (14)

is certainly stable for all n ≥ (k + ℓ). We say certainly because in some cases premature stability may occur for values of n < (k + ℓ).

One observes that the power 3 plethysms for the two fundamental irreducible representations stabilise at n = 3 which is consistent with the stabilsation of the products < s; (0) > × < 1; (μ) > and < s; (1) > × < 1; (μ) > at n = 3 and for similar reasons stabilsation of power N plethysms must occur at n = N as observed. Again, premature stabilisation for individual plethysms may occur for n < N. Thus foe N = 3 all the plethysms stabilise at n = 2 except for s_{1^{3}}(< s; (1) >) which stabilises at n = 3. Stabilisation for arbitrary N occurs at n = N − 1 except for s_{1^{N}} (< s; (1) >) which stabilises at n = N.

Plethysms and conjugacy mappings Below we give two short examples of plethysms with terms kept to weight 10.

s_{4}(< s; (0) >) = < 2; (0) >  + < 2; (4) >  + < 2; (4^{2}) > + < 2; (6) >
+ < 2; (62) > + < 2; (73) > +2 < 2; (8) > + < 2; (91) >
+ < 2; (10 >

\begin{aligned}s_{1^{4}}(< s; (1) >) = & < 2; (1^{4}) > & + < 2; (41^{2}) > & + < 2; (4^{2}) > & + < 2; (61^{2}) > \\ & + < 2; (62) > & + < 2; (73) > & +2 < 2; (81^{2}) > & + < 2; (91) > \end{aligned}

Looking at the above results one cannot help but be struck by the apparent simple mapping between them. Indeed looking at much more extensive tabulations one observes that the terms in s_{λ}(< s; (0) >) are simply related to those of sλ̃(< s; (1) >) by a one-to-one mapping subject to the following simple rules:-

λ ⊢ 2  (0) → (1^{2})
λ ⊢ 3  (0) → (1^{3}) (a) → (a1) (a1) → (a)
λ ⊢ 4  (0) → (1^{4}) (a) → (a1^{2}) (a1^{2}) → (a)
λ ⊢ 5  (0) → (1^{5}) (a) → (a1^{3}) (ab) → (ab1) (ab1) → (ab)
λ ⊢ 6  (0) → (1^{6}) (a) → (a1^{4}) (a1^{4}) → (a) (ab) → (ab1^{2}) (ab1^{2}) → (ab)

The explanation of such simple results remains unknown and deserves further study.

Concluding remarks

The study of plethysms for the non-compact group Sp(2n, R) throws up many surprises that could be of interest to combinatorialists. The study of plethysms for other non-compact groups, such as SO(4, 2) which plays a key role in Coulomb systems, is completely unknown. I hope in these notes I might stimulate others to consider some of the problems raised herein. Acknowlegements

This work has been supported by a grant from the Polish KBN. All calculations were made using SCHUR^{18}. I thank the seminar organisers for the opportunity to participate in a stimulating and well organised meeting.

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18. See http://smc.vnet.net/Christensen.html or http://www.phys.uni.torun.pl/∼bgw/